In Section  we study the basic properties of \(\Gamma \) . First, we observe that the contact surgery graph \(\Gamma \)  is connected by the work of Ding–Geiges [1]} who showed that any contact 3-manifold can be constructed from the standard tight contact structure \(\xi _{\mathrm {st}}\)  on \(S^3\)  by a sequence of \((\pm 1)\) -contact surgeries. On the other hand, we know from [2]}, [3]} that a contact manifold \((M,\xi )\)  is Stein fillable if and only if there exists a directed path from \(v_{(\#_k S^1\times S^2,\xi _{\mathrm {st}})}\)  to \(v_{(M,\xi )}\)  and thus \(\Gamma \)  is not strongly connected since there exists non-Stein fillable contact manifolds. (Recall that an oriented graph is strongly connected if for any pair of vertices \((v_1,v_2)\)  there exist a path from \(v_1\)  to \(v_2\)  following the orientations of the edges.) Etnyre–Honda [4]} showed that there exists a directed path from any vertex corresponding to a given overtwisted contact manifold to any other vertex, i.e. \(\Gamma \)  is strongly connected from any vertex corresponding to an overtwisted contact manifold. The question if there exists a vertex such that \(\Gamma \)  is strongly connected to that vertex is equivalent to the open question if there exists a maximal element with respect to the Stein cobordism relation [5]}.
